4 Answers
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JK Rowling has specifically said that the Nicholas Flamel in Harry Potter is the real person:

Nicholas Flamel is in the book 'the Da Vinci Code', did you get his name from there?

No, Nicholas Flamel is a historical character. Flamel lived in France in the fourteenth century and is supposed to have discovered how to make a philosopher's stone. There are mentions of sightings of him through the centuries because he was supposed to have gained immortality. There are still streets named after Flamel and his wife Perenelle in Paris.(old jkrowling.com)

Nicolas Flamel was a real person. I read about him in my early twenties when I came across one of the versions of his life story. It told how he had bought a mysterious book called The Book of Abraham the Jew, which was full of strange symbols and which Flamel realised were instructions on alchemy. The story went that he subsequently made it his life's work to produce the Philosopher's Stone.(Pottermore - Nicholas Flamel)

She has also said (while under oath) that Paracelsus is a real person.

“I would not go that way if I were you,” said Nearly Headless Nick, drifting disconcertingly through a wall just ahead of him as he walked down the passage. “Peeves is planning an amusing joke on the next person to pass the bust of Paracelsus halfway down the corridor.”
“Does it involve Paracelsus falling on top of the person’s head?” asked Harry.(Harry Potter and the Order of the Phoenix - Chapter 14)

Many other names of real historical people can be found on the chocolate frog cards:

“Oh, of course, you wouldn’t know — Chocolate Frogs have cards, inside them, you know, to collect — famous witches and wizards. I’ve got about five hundred, but I haven’t got Agrippa or Ptolemy.” (...) he had not only Dumbledore and Morgana, but Hengist of Woodcroft, Alberic Grunnion, Circe, Paracelsus, and Merlin.(Harry Potter and the Philosopher's Stone - Chapter 6)

Q. Let's put on the screen Exhibit 34. Can you please tell the Court what Exhibit 34 is.
A. This is a list of the famous wizards, well, fictional famous wizards, and their achievements and dates of life and death -- of birth and death, that I provided to Electronic Arts.
Q. Did you make these wizards up?
A. I did. Occasionally there is someone who existed in reality. I've taken some liberties with their biography.(JK Rowling's testimony at Warner Bros vs RDR Books)

The Pottermore article on "Minsters of Magic" refers extensively to real life politicians, notably Lord North, King George III, Queen Victoria, Lord Palmerston and Winston Churchill

Leonard Spencer-Moon1939 - 1948
A sound Minister who rose through the ranks from being tea-boy in the Department of Magical Accidents and Catastrophes. Oversaw a great period of international wizarding and Muggle conflict. Enjoyed a good working relationship with Winston Churchill.

In July 1999, nine-year-old Natalie McDonald, a Harry Potter fan dying
of leukemia, sent a letter that arrived just after Rowling had gone on
a two-week vacation. When she returned, Rowling quickly sent e-mails
to Natalie and her mother, Valerie. Although Natalie died the day
before, Joanne and Valerie struck up a correspondence, and as a
tribute to Valerie's daughter, the Sorting Hat sends Natalie McDonald
to Gryffindor, making Natalie the only "real" person to appear so far
in the Harry Potter series (Goblet of Fire 159).

@DVK You know, that's like figuring out Uruk and Shuruppak were real places (was that known before the Epic of Gilgamesh was found?) and using that to justify the existence of Humbaba (who, even in-story, wasn't from anywhere near).
– January First-of-MayJan 16 '16 at 15:23

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@JanuaryFirst-of-May - not quite the same. Humbaba is clearly a made up monster. Circe may very well have been some sort of a queen (sorceress, obviously no, but neither was Nicholas Flamel :) . Mind you, I'm not arguing that Circe was real, merely that there's a reasonable plausibility to consider the idea. "may have... prototypes".
– DVK-on-Ahch-ToJan 16 '16 at 15:49

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@JanuaryFirst-of-May - "there should be a separate "characters from real-world legends" bunch" - yes there should be. It's called "figures that are more mythological" in my answer :)
– DVK-on-Ahch-ToJan 16 '16 at 22:01

Apparently, many real people are mentioned on various book covers, film posters, and Muggle newspapers throughout the last several movies (though many of the HBP mentions are chronologically inappropriate for the 1990s), a few come up in Beedle the Bard (I do not know the details), and a few are on Famous Wizard Cards (Leonardo da Vinci is once that didn't come up in the previous answers). Many others are mentioned in Pottermore articles. A few appear on portraits (such as Thomas More).

Nicholas and Perenelle Flamel are apparently (at least, according to the Wiki) the closest you get to actual people being mentioned in-story.